MOX–Report No. 59/2013

One-dimensional surrogate models foradvection-diffusion problems

Aletti, M.; Bortolossi, A.; Perotto, S.; Veneziani,A.

MOX, Dipartimento di Matematica “F. Brioschi”Politecnico di Milano, Via Bonardi 9 - 20133 Milano (Italy)

mox@mate.polimi.it http://mox.polimi.it

One-dimensional surrogate models for

advection-diffusion problems

Matteo Aletti, Andrea Bortolossi,

Simona Perotto♯, and Alessandro Veneziani†

November 20, 2013

♯ MOX– Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”

Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

simona.perotto@polimi.it, matteo.aletti@mail.polimi.it,

andrea.bortolossi@mail.polimi.it

† Department of Mathematics and Computer Science, Emory University400 Dowman Dr., 30322, Atlanta, GA, USA

ale@mathcs.emory.edu

Keywords: model reduction, geometrical multiscale, modal expansion, finiteelements

AMS Subject Classification: 65N30, 65T40

Abstract

Numerical solution of partial differential equations can be made moretractable by model reduction techniques. For instance, when the problemat hand presents a main direction of the dynamics (such as blood flow inarteries), it may be conveniently reduced to a 1D model. Here we comparetwo strategies to obtain this model reduction, applied to classical advection-diffusion equations in domains where one dimension dominates the others.

1 Introduction

Many applications in scientific computing demand for surrogate models, i.e.,simplified models which are expected to be computationally affordable and reli-able from a modeling viewpoint. Problems presenting an evident main direction,such as blood flow in arteries, gas dynamics in internal combustion engines, etc.,are naturally reduced to 1D equations along the coordinate of the main (or“axial” as opposed to “transverse”) direction. Here we consider and comparetwo different strategies to get surrogate models for this kind of problems. The

1

first procedure stems from an appropriate average of the equation along thetransverse direction, combined with (plausible) problem-dependent simplifyingassumptions. The second approach comes from a different representation of theaxial and of the transverse dynamics, according to what has been called a Hi-erarchical Model (Hi-Mod) reduction [4]. In particular, transverse dynamics arerepresented by a modal expansion, that is supposed to require just a few modesfor the nature of the problem. This leads to solve a system of 1D coupled equa-tions. At the bottom line, when using just one transverse mode, this leads to agenuinely 1D model. For the sake of comparison, these two reduction proceduresare applied to the following two-dimensional advection-diffusion problem

−µ∆u + b · ∇u = f in Ω ≡ (0, L) × (−R0, R0)

u = g on Γin ≡ 0 × (−R0, R0)

µ∂u

∂n= 0 on Γout ≡ L × (−R0, R0)

µ∂u

∂n+ χu = uext on Γlat ≡ ∂Ω\(Γin ∪ Γout),

(1)

where x is the main direction, y is the transverse one, and with µ ∈ L∞(Ω), b =(b1, b2)

T ∈ [W 1,∞(Ω)]2, f ∈ L2(Ω), g ∈ H1/2(Γin), χ ∈ L∞(Ω), uext ∈ L2(Γlat),µ∂u/∂n the conormal derivative of u. Standard notation are adopted for theSobolev spaces. We distinguish in the domain Ω a supporting fiber Ω1D alignedwith the main stream and a set of transverse fibers γx, with x ∈ Ω1D, parallel tothe secondary transverse dynamics. Since Ω coincides with a rectangle, γx = γ,for each x.We assume suitable assumptions on the data to guarantee the well-posedness ofthe weak form of (1), i.e.,

find u ∈ V ≡ H1Γin

(Ω) s.t. a(u, v) = F (v) ∀v ∈ V, (2)

with a(u, v) =∫Ω[µ∇u · ∇v + b · ∇uv] dΩ +

∫Γlat

χuv dΓ and F (v) =∫Ω

fv dΩ +∫Γlat

uextv dΓ − a(ρg, v), ρg denoting a lifting of g on Γin. Problem (1) models,for instance, the oxygen transport inside an artery. In this case, u representsthe oxygen partial pressure, µ denotes the diffusivity of oxygen in blood, fieldb takes into account the blood dynamics, f is a generic sink or source term, gusually coincides with a concentration profile, the Robin boundary conditionsmodel the absorption of the oxygen through the vessel walls, with χ dependingon the absorption properties of the wall and uext measuring the oxygen partialpressure outside the vessel. For simplicity, we assume µ and χ constant.When comparing the two approaches mentioned above, we address in particularthe combination of models with different accuracy. For the first approach, thisleads to what has been called a geometrical multiscale formulation [2]. For Hi-Mod reduction, this is obtained by selecting a different number of modes indifferent regions of the domain [4, 5].

2

2 A transverse average model

We particularize the approach in [3] for modeling the transport of solutes inarteries with bifurcations to an elliptic setting. Let us introduce the transverseprofile of the solution, given by

p(x, y) =u(x, y)

U(x)with U(x) =

1

|γ|

∫

γu(x, y) dy,

U(x) denoting the mean of the solution along the transverse (constant) section γof Ω. As first modeling hypothesis, we assume that the profile p does not dependon x, i.e., only the mean of the solution may vary along the x-direction. Thus,after separation of variables, the solution u can be regarded as a certain profilevarying in y tuned by a function varying along x, i.e., u(x, y) = U(x)p(y). Byexploiting this representation of u in the assignment of the boundary conditionson Γlat, i.e., on the boundary of γ, we get

(± µ

∂p(y)

∂y

)∣∣∣y=±R0

=(− χp(y) +

uext(x)

U(x)

)∣∣∣y=±R0

.

Consistently with the previous assumption on p, we postulate that the ratiouext(x)/U(x) is constant along the whole length of the domain. Finally, weconstrain the advective field, by assuming ∇ · b = 0 and b|Γlat

= 0. Since b

is divergence-free, we can rewrite the full model (1) in a conservative form, as−µ∆u + ∇ ·

(bu

)= f . Now, integrating with respect to y along γ, we obtain

− µ∂2

∂x2

∫

γu(x, y) dy − µ

∂u(x, y)

∂y

∣∣∣y=R0

y=−R0

+∂

∂x

∫

γ

[b1(x, y)u(x, y)

]dy

+(b2(x, y)u(x, y)

)∣∣∣y=R0

y=−R0

=

∫

γf(x, y) dy.

By exploiting the Robin conditions and the hypothesis on b|Γlat, we have

− µ∂2

∂x2

∫

γu(x, y) dy + χ

(u(x, R0) + u(x,−R0)

)+

∂

∂x

∫

γ

[b1(x, y)u(x, y)

]dy

=

∫

γf(x, y) dy + uext(x, R0) + uext(x,−R0).

Now, we exploit the factorization u(x, y) = U(x)p(y) assumed for the solution utogether with the fact that, by definition, the mean of p along γ is equal to one,to get the desired averaged 1D model (the primes denoting x-differentiation)

−µU ′′(x) + (U(x)wr(x))′ + σrU(x) = fr(x) for x ∈ (0, L), (3)

with

wr(x) =1

|γ|

∫

γb1(x, y)p(y) dy, σr = χ

p(R0) + p(−R0)

|γ|,

fr(x) =1

|γ|

∫

γf(x, y) dy +

uext(x, R0) + uext(x,−R0)

|γ|.

(4)

3

The reduction procedure leads from a 2D advection-diffusion problem to a 1Dadvection-diffusion-reaction problem. To close the model we need to select aprofile p in (4). For simplicity, it may be assumed constant or, more in general, itis suggested by physical considerations. It could be advantageous an automaticcriterion to select p. A strategy in such a direction is proposed in the nextsection.

Remark 2.1 From a physical viewpoint, the most restrictive hypothesis for de-riving model (3) is the independence of p on x. Nevertheless, the numericalvalidation shows that this surrogate model provides reliable results even whenthis hypothesis is not strictly guaranteed. The second assumption is reasonable,at least in haemodynamics, since the ratio uext(x)/U(x) may be reliably con-sidered constant. The two requirements on b are standard in a haemodynamiccontext. Hypothesis ∇ · b = 0 ensures the incompressibility of the blood, whileassumption b|Γlat

= 0 imposes a no-slip condition on Γlat.

2.1 A geometrical multiscale approach

A geometrical multiscale formulation consists of coupling dimensionally hetero-geneous models. The idea is to alternate a full-dimensional model with suitabledownscaled models to be associated with the areas characterized by the mostcomplex and by the simplest dynamics, respectively (see, e.g., [2, Chapter 11]).The identification of appropriate matching conditions and the location of the in-terface between the two models represent the main issues of this approach. Weidentify the full model with (1) and the downscaled model with (3). We chooseΩ = (0, 10)×(0, 1), µ = 1, b = (20, 0)T , f = 10

((x−1.5)2+0.4(y−0.5)2 < 0.01

),

χ = 1 and uext = 0.02. We assign a homogeneous Neumann condition onΓout = 10 × (0, 1) and a profile compatible with the conditions along Γlat

on Γin. In Figure 1 (top-left), we provide the contour plots of the full solu-tion approximated via linear finite elements on a uniform unstructured grid of8918 elements. The solution exhibits more significant transverse dynamics inthe leftmost part of the domain, where the source term is localized. Conversely,the solution profile is less fluctuating in the rightmost part of Ω, as assumedin the derivation of model (3). This suggests to split Ω into two subdomains,Ω1 and Ω2, such that Ω = Ω1 ∪ Ω2. On Ω1 we solve problem (1), while weresort to (3) in Ω2. Both the problems are discretized via linear finite elementson uniform meshes. The coupling between the two models is performed via arelaxed Neumann/Dirichlet scheme. In more detail, we exploit the derivativeof the 1D surrogate solution u2 to assign a constant Neumann condition on Ω1

as µ∂u1/∂n(xi, y) = u′2(xi), where u1 is the full solution defined on Ω1 and

Γxi= xi × (0, 1) identifies the interface Ω1 ∩ Ω2. To correctly define the

problem on Ω2, we have to properly select the boundary condition at xi andthe solution profile. As Dirichlet data we assign u2(xi) = |γ|−1

∫γ u1(xi, y) dy,

while we follow a new approach to select p(y) at xi. The idea is to exploit

4

0 2 4 6 8 100.000.250.500.751.00

0 2 4 6 8 101.1

1.4

1.7

2.0

0 2 4 6 8 100.000.250.500.751.00

0 2 4 6 8 100.000.250.500.751.00

Figure 1: Geometrical multiscale: full solution (top-left); graph of σr (top-right);coupled solution for Γ4 (bottom-left) and Γ2.5 (bottom-right)

the problem in Ω1 instead of resorting to an a priori selection. Thus, we pickp(y) = |γ|u1(xi, y)/

∫γ u1(xi, y) dy. This definition justifies the prescription of

a Neumann condition on the left hand side of Γxito allow the solution profile

to develop freely. Indeed, the adoption of the surrogate model in Ω2, implicitlyassumes that p is completely developed at Γxi

. Figure 1 (bottom) comparestwo couplings associated with different interfaces, i.e., Γ4 and Γ2.5, respectively.The second choice introduces the interface where the transverse dynamics arestill too significant, thus violating the hypothesis on a fully developed profile.We provide a bidimensional visualization also for the surrogate model simply byusing relation u(x, y) = U(x)p(y). In Figure 1 (top-right) we show the reactivecoefficient in (4), computed via the profile of the full solution. Since σr stronglydepends on p, we argue that when the profile stabilizes, σr reaches a constantvalue. So a possible heuristic way to select Γxi

is to locate it in a region whereσr is constant.

3 Hi-Mod reduction

Hi-Mod reduction is an alternative approach to “compress” high dimensionalproblems. In this case, a full 2D (or even 3D) model is reduced to a system of1D coupled differential problems associated with the dominant dynamics [4]. Inthe geometric setting of a Hi-Mod formulation, for any x ∈ Ω1D, we introducea map ψx between the generic fiber γx and a reference fiber γ, so that the com-putational domain Ω is mapped into the reference domain Ω = Ω1D × γ via themap Ψ, given by Ψ(z) = z, where z = (x, y) ∈ Ω, z = (x, y) ∈ Ω, with x = x andy = ψx(y). In particular, for the domain Ω in (1) a unique map ψ can be usedfor each point x ∈ Ω1D. The Hi-Mod approach strongly relies upon the fiberstructure postulated on Ω. The idea is to differently tackle the dependence of

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the full solution on the dominant and on the transverse directions. We performa modal approximation of the transverse dynamics coupled with a Galerkin rep-resentation along the axial direction. The rationale driving this approach is thatthe transverse dynamics can be suitably described with a few degrees of (modal)freedom, resulting in a hierarchy of one-dimensional models which differ eachother according to the number of included transverse modes. To state the Hi-Mod reduced formulation for problem (1), we move from the weak form (2). Now,let V1D be a space spanned by functions defined on Ω1D which properly includesthe boundary conditions assigned along Γin and Γout, and let ϕkk∈N

+ be a

modal basis of functions in H1(γ), orthonormal with respect to the L2(γ)-scalarproduct and compatible with the boundary conditions along Γlat. As a conse-quence, we look for a reduced solution um which belongs to the Hi-Mod reducedspace Vm =

vm(x, y) =

∑mk=1

vk(x)ϕk(ψ(y)), with vk ∈ V1D, x ∈ Ω1D, y ∈ γ.

A conformity and a spectral approximability hypothesis are introduced on Vm

to guarantee the well-posedness and the convergence of um to u [4]. We identifythe Galerkin representation along Ω1D with a finite element discretization, sothat the modal coefficients belong to a finite element space V h

1D ⊂ V1D asso-ciated with a partition Th of Ω1D. Thus, the Hi-Mod reduced form for (2) is:for a certain modal index m ∈ N

+, find uhk ∈ V h

1D, with k = 1, . . . , m, suchthat

∑mk=1

a(uhk ϕk, θiϕj) = F (θiϕj), with j = 1, . . . , m and i = 1, . . . , Nh,

where θi denotes the generic finite element basis function in V h1D and with

Nh = dim(V h1D) < +∞. From a computational viewpoint, the Hi-Mod for-

mulation leads to solve a system of m coupled 1D advection-diffusion-reactionproblems instead of problem (1). As in the derivation of the surrogate model (3),the Hi-Mod reduction procedure yields reactive terms, while no reactive contri-bution is included in the full model. The system is characterized by an m × mblock matrix, where each block is an Nh × Nh matrix exhibiting the sparsitypattern typical of the selected finite element space.The modal index m can be selected a priori moving from some preliminaryknowledge of the phenomenon at hand [4] or automatically, driven by an a pos-teriori modeling error analysis [5]. Another important issue is the choice ofthe modal basis, in particular when Robin boundary conditions are assignedon Γlat as in (1). We build a specific modal basis able to automatically in-clude these conditions. The idea proposed in [1] is to solve on γ an auxiliarySturm-Liouville eigenvalue problem, with conditions on ∂γ coinciding with theconditions assigned on Γlat. We call this modal basis educated basis.

3.1 Piecewise Hi-Mod reduction

Now, the idea is to properly exploit the hierarchy of models provided by theHi-Mod reduced space to couple models with a different accuracy. A differentchoice for the modal index m identifies a reduced model with a certain level ofdetail in describing the phenomenon at hand. As a consequence, by properlytuning m over different regions of Ω, we are able to capture the local significant

6

0 2 4 6 8 100.00.20.40.60.81.0

0 2 4 6 8 100.00.20.40.60.81.0

Figure 2: Piecewise Hi-Mod reduction: reduced solutions associated with 5, 2modes, the interface is Γ4 (on the left) and Γ2.5 (on the right); h = 0.05 on Ω1D

features of the solution with a relatively low number of degrees of freedom.Following [4], we denote this approach by piecewise Hi-Mod reduction. Thisleads to dimensionally homogeneous models (yet with a locally varying level ofaccuracy), as opposed to the geometrical multiscale approach. For instance,with reference to the test case in Figure 1, we can preserve the two splittingsof the domain identified by Γ4 and Γ2.5 and employ a number of modes in Ω1

higher than in Ω2, e.g, 5 and 2, respectively. This choice is motivated by thefact that the most complex dynamics are localized in Ω1 and, consequently,more modes are demanded in this area. To glue the two models we employ arelaxed Neumann/Dirichlet scheme as in the geometrical multiscale formulation.At each iteration of this scheme, we apply a uniform Hi-Mod reduction on Ω1

and Ω2, separately, i.e., we solve two systems of coupled 1D problems with ablock matrix of order 5N1

h and 2N2h , respectively N i

h denoting the dimension ofthe one dimensional finite element space introduced on Ω1D ∩ Ωi, for i = 1, 2.As detailed in [5], to rigorously formalize the piecewise Hi-Mod approach, weintroduce a suitable broken Sobolev space, endowed with an integral conditionwhich weakly enforces the continuity of the reduced solution in correspondencewith the minimum number of modes common on the whole Ω. This does notnecessarily guarantee the conformity of the piecewise reduced solution. Thisis evident in Figure 2. The loss of conformity is particularly significant whenthe interface is located in an area involved by strong transverse dynamics. Thereduced solution in Figure 2 (left) is in good agreement with the full one inFigure 1 (top-left) and it is very similar to the one in Figure 1 (bottom-left).

4 A numerical comparison

For a fair comparison between the two approaches, we consider here a test casewhere the “low-fidelity” model (the genuine 1D in the geometrical multiscaleand a low-mode approximation in the Hi-Mod) are straightforwardly compara-ble. This means that we employ a single mode in the Hi-Mod approximation. Inparticular, we consider problem (1) with Ω = (0, 6)× (0, 1), µ = 1, b = (20, 0)T ,f = 10

([(x−1.5)2 +0.4(y−0.75)2 < 0.01]+ [(x−1.5)2 +0.4(y−0.25)2 < 0.01]

),

χ = 3 and uext = 0.05. The boundary conditions are as in Section 2.1 and

7

s

0 1 2 3 4 5 60.00.20.40.60.81.0

0.060.090.120.15

0 1 2 3 4 50.00.20.40.60.81.0

0.060.090.120.15

0 1 2 3 4 5 60.00.20.40.60.8

0.060.090.120.15

0 1 2 3 4 5 60.00.20.40.60.8

0.060.090.120.15

Figure 3: Full solution (top-left); geometrical multiscale solution (top-right) andHi-Mod solution with 3, 1 (bottom-left) and 5, 1 (right) modes

the interface is located at x = 3. Numerical results are provided in Figure 3.The top-left panel displays the full solution discretized via standard linear finiteelements on a uniform unstructured grid of 5084 triangles. The top-right panelshows the geometrical multiscale solution. This is fairly accurate even though itsuffers from an underestimation of the reactive term induced by the 1D average.This is evident in the contour line associated with the value 0.09. The bottompanels display the Hi-Mod solution, having a “low-fidelity” model with m = 1and two different models for the “high-fidelity” part. In particular, on the leftwe take m = 3 which is clearly not enough to capture reliably the solution inthe leftmost domain. In the right-panel, with m = 5 we have a pretty accuratesolution, where the inaccuracy present in the geometrical multiscale solution aswell as the model non-conformity do not pollute significantly the results.An extensive comparison between the two approaches cannot be clearly com-pleted by these preliminary results. As a matter of fact, the computationaladvantages of the one approach over the other must be evaluated on 3D morerealistic test cases, solved with compiled softwares. However, we may noticethat, even though the Hi-Mod approach relies entirely on a “psychologically”1D representation of the solution within a dimensionally heterogeneous frame-work, it may provide accurate solution also in presence of significant transversecomponents. For this reason, we do expect it may lead to easily implementedand manageable solvers, with competitive performances in terms of both ac-curacy and efficiency. A framework of investigation of practical interest is theblood flow simulation in a network of arteries.

References

[1] Aletti, M., Perotto, S., Veneziani, A. In preparation (2013)

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[2] Formaggia, L., Quarteroni, A., Veneziani, A. eds., Cardiovascular Mathe-matics, Modeling, Simulation and Applications 1, Springer-Verlag, Berlin,Heidelberg, 2009.

[3] Krpo, A.: Modelisation par des modeles 1D de l’ecoulement du sang et dutransport de masse dans des arteres avec bifurcation. Master Thesis, EcolePolytechnique Federale de Lausanne (2005)

[4] Perotto, S., Ern, A., Veneziani, A.: Hierarchical local model reduction forelliptic problems: a domain decomposition approach. Multiscale Model.Simul. 8(4), 1102–1127 (2010)

[5] Perotto, S., Veneziani, A.: Coupled model and grid adaptivity in hierarchi-cal reduction of elliptic problems. J. Sci. Comput. in press (2013)

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